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Technically --you dont have a flasher.....The 4 way button on the dash
is a combination turn/ hazard flasher control "MODULE"...

This is a common problem with these switches. I have replaced several of
them over the last few months. The switch can be replaced without all
of the disassembly that is recommended. I have done it. There is a screw
holding the switch into the dash behind the bezels, but it is possible
that it can be done without spending hours on it. The Grand Am/Prix cars
were all the same, just a slightly different design.

Replacement
The turn and hazard flashers are integrated into one flasher on these models. The flasher is located in the hazard warning switch and the hazard warning switch is located on the trim plate. Replace the flasher by unplugging the old one, removing it from the panel, and plugging in the new one.

Remove the bezel from the ignition switch.

Remove the bezel from the ignition switch

Carefully remove the instrument panel accessory trim plate.

To remove the trim plate, use a suitable prytool to carefully apply pressure at the retaining clip points . . .

. . . then gently pull the accessory trim plate away

Detach the electrical connector from the hazard warning switch.

Unplug the electrical connector from the hazard warning switch . . .

Unfasten the screw securing the hazard warning switch and remove the switch.

. . . then unfasten the screw securing the hazard warning switch and remove the switch from the vehicle

The flasher is located above the fuse panel. The relays are built into the switch itself. They have a recall on the 4 way flasher switch because it is faulty and causes blinkers not to turn on. Very common Malibu and other Gm problem

The flasher unit has gone bad and needs to be changed. The flasher module is a part of the hazard module also. The clicking is the unit is still working but there is not enough voltage to activate the lights to flash. It's common on the Malibu's and other GM product cars similar to the Malibu like Grand Am's Ect.
I believe Auto Zone has the replacement part for you car and is more reasonable in price then the dealer and the y also have the step by step instructions to replace the flasher module.
Good luck and keep me posted, be glad to help.